We don't put anything down--if she has a placemat, there is too good of a chance of her pulling it off, along with everything on it (plate, utensils, cup, as well as the mess of food). In fact, we had to stop using a tablecloth or placemats in general when she was really little because it was just too messy/dangerous.
So we just eat at the table, wipe it down afterwards--it cleans up really easily. The only part I get tired of is needing to sweep and/or mop after every meal. I usually only manage to do so once a day, unless there is a particularly big mess.
~Kim